Chromatic number and subtrees of graphs (Q1692708)

A. Gyárfás and D. Sumner independently conjectured that for every tree \(T\) the family of graphs not inducing \(T\) as a subgraph is \(\omega\)-bounded, where \(\omega\) denotes the chromatic number. There have been many papers on this still unsolved conjecture. The authors prove that trees obtained by identifying one end of a path with the central vertex of a star are \(\omega\)-bounded. They also improve the bound on a topological version of the conjecture.